P-adaptation of successive correction k-exact finite volume schemes for compressible flows

A $p$-adaptation strategy is developed in the framework of successive correction $k$-exact finite volume schemes. A new adaptation indicator based on the decay of the successive correction terms used to reconstruct the solution within one cell is introduced to drive the adaptation process. The criterion relies on low-order derivatives, is efficiently estimated as part of the successive correction process, and identifies well flow regions characterized by steep gradients. Unlike other strategies in the literature, $p$-adaptation is only used to increase solution accuracy, while robust slope limiters are used to control the appearance of spurious oscillations. The performance of the proposed adaptive method is evaluated for a variety of 2D steady and unsteady, inviscid and viscous compressible flow configurations, as well as for a 3D transonic wing. The results show the effectiveness of $p$-adaptivity in achieving high-order solution quality while maintaining the computational effort close to that of a second-order (one-exact) scheme in terms of memory load and computation time.